pureM

Logic

Ex1

What reasonable conclusion can you make?

A : All Lower 6 students take Liberal Studies.
B : You are wrong.

Ex2

Conditionals "if p, then q"

Determine the truth value (true/false) for each of the following statements.

					Truth value
	If p, then q.		p	q	If p, then q
(i)	If 2 is even, then 20 is odd.		T	F	?
(ii)	If 1 = 2, then 3 = 3.*		F	T	?
(iii)	If 1 = 2, then 11 = 22.**		F	F	?

	Possible explanations for
*(ii)	If 1 = 2, then 2 = 1. Adding the two,	1+2=2+1, 3=3!
**(iii)	If 1 = 2, then 10=20. Adding the two,	11=22!

	An example for (iii) in daily life: If a question in the examination is wrong, even if the candidate makes a logically sound argument, the candidate probably will reach a wrong conclusion.

Ex3

(a)

The box is cubic in shape and it is red in colour. Negation of this statement is

Guess a formula for the negation of "p and q".

(b)

The printer is not working. The connecting cable to the computer is loose or the printer is out of order. Negation of this statement is

Guess a formula for the negation of "p or q".

(c)(i)

If it is a Tuesday, then John needs to go to school. Negation of this statement is

(Hint: Note that "it is a Tuesday" is given. Only the conclusion "John needs to go to school" should be discussed. )

(c)(ii)

Amy reached the final round of the TV game show. Before her were 3 keys from which she would choose one to open the door that leaded to the grand prize. Negation of the statement "If Amy chooses key #2, then she will win" is (Again note that "Amy chooses key #2" is a given fact, discuss only the conclusion.)

Can you guess a formula for the negation of "If p, then q"?

Go back to the notes and look up the correct formulae for the

negation of

(a) p and q

(b) p or q

Do they agree with your intution?Now prove these formulae mathematically by completing the truth tables of (a) and (b).

You will find these formulae very useful in logical deduction, in mathematics as well as in daily life experience. Try Ex 4 using these formulae.

Ex5

There are 3 balls in a box.

(a)

All the balls are blue in colour. Negation of the statement is

(b)

Some balls are red in colour. Negation of the statement is

Go back to the notes and try Ex 5 and Ex6.

Ex6

Hints for Ex6(a) and (*) are given below.

(a)

Statement: All mathematics problems are easy.

Negation of the statement :

in symbol:

equivalent to

in words : Some mathematics problems are not easy.

Also,

Ex7(a)

Consider the obvious fact "If ABCD is a square, then ABCD is a parallelogram".

What words are missing below? Check your answer by clicking on the buttons.

(a) "ABCD is a square" is a ________condition for "ABCD is a parallelogram"
(b)"ABCD is a parallelogram" is a ________condition for "ABCD is a square"
(c)ABCD is a square _____ ABCD is a parallelogram.

Go back to the notes and try Ex7(b).

Ex8

Prove that "If x² is not an even integer, then x is not an even integer."

	Given that x² is not an even integer,
	To show that x is not an even integer.
Pf	*Assume* that *x is an even integer*.
	It follows that x² is an even integer.
	But it is given that x² is not an even integer!!! Contradiction!
	The *assumption* must be incorrect. Hence x is not an even integer.

Ex9

Prove that

is irrational.

Pf	*Assume* thatis *rational*.
	Let =p/q where p and q are integers with no common factor other than 1 and q is not equal to 0.
	By squaring and rearranging terms, we have p² = 2 q² .
	Since p² is even, it follows that p is even.
	Let p = 2m where m is an integer. Subsituting back into the equation, we have (2m)² = 2 q² . On further simplification, 2 m² = q² .
	Since q² is even, it follows that q is even.
	As p and q are both even, then 2 is a common factor of p and q!!! Contradiction!
	The *assumption* must be incorrect. Hence is irrational.

Problem solving

	A power point file on problem solving strategies and presentation techniques

Calculus
	Functions with very special properties (1) A plane filling curve - e.g. Peano curve (2) A function which is not continuous at any point - Dirichlet function (2) A continuous but nowhere differentiable function - Every differentiable function is continuous on IR. The converse is not true. There exist continuous functions on IR which is not necessarily differentable on IR. Actually, there exists a continuous function which is nowhere differentiable(e.g. Weiserstrass function).
Websites
Calculus Toolkit	MathServe Project by Vanderbilt University, USA offers useful calculations
	Limits;Differentiation;Integration;Sum of a series;Partial Fractions;Finding roots...
Curve Sketching	by Vanderbilt University, USA
	sketches curves y= f(x), Parametric Functions,Implicit Functions
	*eg y = sin(3x)+(cos(2x))/6, y = 4x^2 - 3x + 6*
Function Plotter	by Maths Online, University of Vienna
Famous Curves	by University of St. Andrews, Scotland
Thinkquest library	extensive material on differentiation and integration
Java Applets	excellent applets byInernational Education Software
Calculus	recommended sites by Math Forum

e and the natural logarithm

Application

(1)

It is known that for the same principal deposited in a bank, the more frequent the compound interest is credited the more one gets in the end. In other words, daily compounding yields more interest than monthly compounding which in turn yields more than yearly compunding. If a bank offers continuous compounding, is it as good as it sounds? Does it mean that the total amount will increase without bound?

(a)

Compare the amount in a bank account at the end of 10 years for a deposit of $P at an interest rate of 6 % compounded (a) yearly (b) half-yearly (c) monthly (d) daily (e) continuously. Answer

In general, find the total amount in the bank account for a deposit of $P at an interest rate r% compounded continuously over n years.

(Hint: assume the interest is compounded n times a year and then take n to infinity. Note that no matter how frequent interest is compounded, the total amount is bounded by the limit value Pe^r%t)

(b)

The rule of 70:

Suppose it takes n years for an amount of money to double when invested at the rate of r% compounded continuously. The rule of thumb states that nr is approximately 70.

The proof is quite short. Try it.

In fact, even if the amount is compounded annually, the rule of thumb still works provided that r% is small compared to 1. Try proving it! (You may use the approximation that ln (1+x) is close to x when x is small.)

(2)

Franklin Benjamin's will

Franklin Benjamin (1706-1790) American printer, author, diplomat, philosopher, and scientist, inventor of the lighting rod and bifocal glasses

Summary

Franklin Benjamin would give 1000 pounds to Boston (and another 1000 to Philadelphia). The plan was to lend money to young apprentices in these cities at 5% interest with the provision that each borrower should return the interest and part of the principal each year. Franklin predicted that if the plan was exceuted without interruption, the sum would reach 131,000 pounds at the end of 100 years, of which 100,000 pounds were to be allocated to public works in Boston and the remaining 31,000 pounds would continue to be lent to young people in the same manner for another 100 years. He predicted that if there was no unfortunate accident to prevent the operation, the sum would be 4,610,000 pounds.

Fact

Though it was not always possible to find as many borrowers in Boston as Franklin had planned, the managers of the trust did the best they could. At the end of 100 years from the reception of the Franklin gift, in January 1894, the fund had grown from 1000 pounds to 90,000 pounds.

Question

In the first 100 years since the will, the original capital had mutiplied about 90 times instead of 131 times Franklin had imagined. What rate of interest, compounded continuously, would have multipled the capital by 90?

(Answer: 4.5%)

System of Linear Equations

Amy, Ben and Calvin play a game as follows. The player who loses each round must give each of the other players as much money as the player has at that time. In round 1, Amy loses and gives Ben and Calvin as much money as they each have. In round 2, Ben loses, and gives Amy and Calvin as much money as they each then have. Calvin loses in round 3 and gives Amy and Ben as much money as they each have. They decide to quit at this point and discover that they each have $24. How much money did they each start with?

Sol

This problem can be solved using the (1) top down strategy or (2) bottom up strategy

Method 1

	Amy	Ben	Calvin
At start	$x	$y	$z
after round 1
after round 2
after round 3

By completing the above table, you should obtain the following equations

x - y - z = 6, 3y - x - z = 12 and 7z - x - y = 24; from which you can solve for x, y and z. Answer

Linear Equation Solver

Method 2

Instead of considering how much money Amy, Ben and Calvin had originally, work backwards from the moment when each of them has $24 each. Complete the following table and see how easily you can reach exactly the same conclusion as in method 1.

	Amy	Ben	Calvin
At the end	$24	$24	$24
before round 3
before round 2
before round 1

	Matrices
	Powerpoint files: Matrix multiplication, cofactor and inverse Examples on linear transformations
	Partial Fractions
	Thinkquest library - excellent website created by pre-collegiate students
	Complex numbers
	Harry found in his attic the old treasure map his great grandma inherited a long long time ago. His great grandma cherished the map over all these years and kept it a secret until she thought Harry was wise enough to recover the treasure and let the whole family enjoy the great wealth. Sail to 114.5^o North, 23.1^oEast, there lies a small island. At the south of the island is a pasture where an oak tree and a pine tree and a gallows stand. Start walking from the gallows to the oak tree, counting carefully the number of steps taken. At the oak tree, turn right through 90 degrees and walk exactly the same number of steps. Make a mark there (P). Go back to th gallows and walk towards the pine tree, again counting the number of steps needed. From the pine tree, turn left through 90 degrees and again walk exactly the same number of steps. Make a mark there (Q). Start digging midway between P and Q. There lies the treasure. Unfortunately, the wood gallows had vanished completely after all these years. On the other hand, the oak and the pine tree have survived numerous storms and become landmarks of the island. "Why didn't great grandma tell me earlier about the map? I could have easily found the treasure back then!" With great disappointment, Harry threw the map into the fire.
	This is a sad story. It is made even sadder by the fact that has Harry known more about mathematics (notably about complex numbers), he would have easily calculated the exact location of the treasure. If only he has paid more attention in class!
Sol	Imagine that the island lies on the Argand plane. The line joining the oak and the pine is the real axis. Without loss of generality, let O be the midpoint of the two trees and the trees' location are represented by the numbers 1 and -1 respectively. The gallows, P and Q are represented by z, p and q in the Argand plane. Can you use the geometry of complex numbers to find p, q and hence z? Answer

Ans	(a)P(1+0.06)^10 (b)P(1+0.03)^20 (c)P(1+0.005)^120 (d)P(1+0.06/365)^3650 (e)limit value of P(1+0.06/n)^10n as n tends to infinity, on simplifying, this is Pe^0.6. Compare the results of (a) and (e). BACK
Ans	Solving the equations, Amy has $39, Ben $21, Calvin $12 before the game starts. BACK
Ans	p = i(z+1) + 1, q = i(1-z)-1, z = i ! In this case, Harry could have measured the distance between the oak and the pine trees (say x m) and then from midway of the two trees, he should walk in the direction making exactly a right angle with the line joining the two trees for x/2 m. There lies the treasure. BACK